### Nuprl Lemma : causal_order_sigma

`∀[T,A:Type].`
`  ∀L:T List`
`    ∀[R:ℕ||L|| ⟶ ℕ||L|| ⟶ ℙ]. ∀[P,Q:A ⟶ ℕ||L|| ⟶ ℙ].`
`      (Trans(ℕ||L||)(R _1 _2)`
`      `` (∀x:A. causal_order(L;R;λi.P[x;i];λi.Q[x;i]))`
`      `` causal_order(L;R;λi.∃x:A. P[x;i];λi.∃x:A. Q[x;i]))`

Proof

Definitions occuring in Statement :  causal_order: `causal_order(L;R;P;Q)` length: `||as||` list: `T List` trans: `Trans(T;x,y.E[x; y])` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  causal_order: `causal_order(L;R;P;Q)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` exists: `∃x:A. B[x]` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s1;s2]` so_apply: `x[s]` and: `P ∧ Q` int_seg: `{i..j-}` subtype_rel: `A ⊆r B` so_lambda: `λ2x y.t[x; y]` cand: `A c∧ B`
Lemmas referenced :  exists_wf int_seg_wf length_wf all_wf le_wf subtype_rel_self trans_wf list_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation_alt lambdaFormation sqequalHypSubstitution productElimination thin cut introduction extract_by_obid isectElimination hypothesisEquality lambdaEquality applyEquality hypothesis natural_numberEquality functionEquality productEquality setElimination rename instantiate universeEquality because_Cache inhabitedIsType functionIsType universeIsType dependent_functionElimination independent_functionElimination dependent_pairFormation independent_pairFormation

Latex:
\mforall{}[T,A:Type].
\mforall{}L:T  List
\mforall{}[R:\mBbbN{}||L||  {}\mrightarrow{}  \mBbbN{}||L||  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[P,Q:A  {}\mrightarrow{}  \mBbbN{}||L||  {}\mrightarrow{}  \mBbbP{}].
(Trans(\mBbbN{}||L||)(R  \$_{1}\$  \$_{2}\$)
{}\mRightarrow{}  (\mforall{}x:A.  causal\_order(L;R;\mlambda{}i.P[x;i];\mlambda{}i.Q[x;i]))
{}\mRightarrow{}  causal\_order(L;R;\mlambda{}i.\mexists{}x:A.  P[x;i];\mlambda{}i.\mexists{}x:A.  Q[x;i]))

Date html generated: 2019_10_15-AM-10_57_41
Last ObjectModification: 2018_09_27-AM-09_52_36

Theory : list!

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